Gamma Function of One Half/Proof 3
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Theorem
- $\map \Gamma {\dfrac 1 2} = \sqrt \pi$
Proof
\(\ds \map \Gamma {\dfrac 1 2}\) | \(=\) | \(\ds \int_0^{\to \infty} t^{-\frac 1 2} e^{-t} \rd t\) | Definition of Gamma Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_0^{\to \infty} u^{-1} e^{-u^2} 2 u \rd u\) | Integration by Substitution, $\map \phi u = u^2$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \int_0^{\to \infty} e^{-u^2} \rd u\) | Linear Combination of Definite Integrals | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_{\to -\infty}^{\to \infty} e^{-u^2} \rd u\) | Definite Integral of Even Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt \pi\) | Gaussian Integral |
$\blacksquare$
Sources
- 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Solved Problems: The Gamma Function: $30$